Question-152147

Question Number 152147 by john_santu last updated on 26/Aug/21

Answered by Ar Brandon last updated on 26/Aug/21

I_{n} = \int_{- 1}^{1} (\sqrt{x^{2} + \frac{1}{n}} - ∣ x ∣) d x

= 2 \int_{0}^{1} (\sqrt{x^{2} + \frac{1}{n}} - ∣ x ∣) d x

= 2 \int_{0}^{argsh (\sqrt{n})} (\frac{1}{n} \sqrt{\sinh^{2} ϑ + 1}) \cosh ϑ d ϑ - 1

= \frac{1}{n} \int_{0}^{argsh (\sqrt{n})} (\cosh 2 ϑ + 1) d ϑ - 1

= \frac{1}{n} {[\frac{\sinh (2 ϑ)}{2} + ϑ]}_{0}^{argsh (\sqrt{n})} - 1

= \frac{1}{n} (\frac{sh (2 argsh (\sqrt{n}))}{2} + argsh (\sqrt{n})) - 1 = 0